Laplace Transformation Technique Research Articles

Analysis of the multi-term fractional Bateman equations in radioactive decay by means of Mikusiński algebraic calculus

Fractional Bateman equations model the concentrations of nuclear species through a radioactive decay chain, by considering non-Markovian behavior. Cruz-López and Espinosa-Paredes (Nuclear Engineering and Technology, 2022) solved the mathematical problem with a single memory coefficient, based on the Laplace-transform technique. Cruz-López, Espinosa-Paredes and François (Computer Physics Communications, 2022) extended the analysis to several fractional orders, with Laplace transforms and series expansions checked by induction. In this paper, we solve such a case of distinct memory effects, via the more general method of Mikusiński operational calculus. We do not distinguish between equal and different decay rates. Multivariate Mittag-Leffler functions appear in the closed-form expression.

Radiation absorption effect on MHD natural convective flow past an impulsively started infinite vertical plate with chemical reaction and thermal radiation

AbstractThe current work is interested on investigating the impacts of thermal radiation, chemical reaction, and absorption radiation of a hydromagnetic convection‐free mass and heat transfer flow in case of an electrically conducting fluid that passes through a vertical plate moving impulsively. The analytical solutions of the governing momentum, energy, and species concentration equations with the initial and boundary conditions are obtained by the Laplace transformation technique. Graphs for fluid characteristics are used to analyze the impact of changing parametric quantities such as M, N, Sc, Kc, Q, Gr, Gm, and t on the temperature, velocity, concentration, and Sherwood number. We derive the engineering curiosity expressions for the Nusselt number and stress, and at the end, we tabulate and discuss the consequences of new parameters. The magnetic field effect and the chemical reaction are seen to diminish the fluid velocity and concentration, respectively, but in contrast, the absorption radiation effect is seen to accelerate both velocity and temperature. It is closely studied that the Nusselt number and skin friction values for hydrogen consistently exceed those for carbon monoxide.

Significance of radiation on the Brinkman-type nanofluid with variable surface velocity embedding with porous matrix

The current flow phenomena based on the flow of water-based nanofluid corresponding to the variable surface velocity. Further, the electrically conducting fluid embedding with porous matrix with an inclusion of thermal radiation enriches the study as well. Because of the varied industrial applications for the use of nanoparticle concentration with radiating heat that is used in cooling process has significant role for the enhancement of heat transport properties. Besides to it in several other applications like engineering, biomedical, and nuclear science the contribution of nanoparticle shows its extensive properties. The main goal of the proposed investigation is to study the influence of thermal radiation on the Brinkman-type nanofluid in the presence of porous matrix. Further, the behavior of variable surface velocity enriches the study. The proposed designed time dependent model associated with various physical restrictions is taken care of analytically using Laplace Transformation Technique and their behavior is deliberated through graphs. However, the validation as well as convergence criteria of the present methodology correlate with the earlier investigation.

A new constitutive theory of nonlocal piezoelectric thermoelasticity based on nonlocal single-phase lag heat conduction and structural transient thermo-electromechanical response of piezoelectric nanorod

Thermo-electromechanical coupling analysis at micro and nano temporal and spatial scales becomes significantly important with extensive application of ultrafast heating technology to micro/nano-machining and manufacturing of piezoelectric micro/nano-devices, where the size effects in heat conduction and elastic deformation remarkably increase. This article aims to establish a new nonlocal piezoelectric thermoelasticity theory based on newly developed model of Eringen-type nonlocal single-phase-lag heat conduction via Eringen’s nonlocal continuum mechanics. The proposed model is applied to analyze the thermo-electromechanical responses of piezoelectric nanorod by Laplace transformation technique. The effects of nonlocal characteristic lengths on wave propagations and structural transient response are evaluated and discussed.

Analysis of Newtonian heating in fractional hybrid nanofluid with the influence of transverse magnetic field

This article deals with the effects of Newtonian heating in second grade Hybrid nanofluid. For memory effects the problem is modeled with the help of generalized Fourier law along with generalized boundary conditions. A hybrid nanofluid model is used with nano particles of the form Magnetic Iron Oxide (Fe3O4) and Silver (Ag) nano particles with engine oil (EO) as based fluid. A uniform magnetic field perpendicular to the plate is also taken. General solutions of the problems are obtained with the help of Laplace Transform Technique. Influence of related physical parameters on the fluid flow are discussed with the help of graphical display of the solutions. It is also observed that generalized boundary conditions provides more effective and new solutions for the oscillating flow of hybrid nanofluid which produces a better way of understanding of hybrid nanofluid.

Vibration of Non-Uniform Bernoulli-Euler Beam Under Moving Distributed Masses Resting on Pasternak Elastic Foundation Subjected to Variable Magnitude

This paper investigates the dynamic response of a clamped-clamped non-uniform Bernoulli-Euler beam resting on a Pasternak elastic foundation to variable magnitude moving distributed masses. The predicament is dictated by a partial differential equation of fourth order, which features coefficients that are both variable and singular. The primary aim is to derive an analytical solution for this category of a dynamic problem. To achieve this, we employ the method of Galerkin with a series representation of the Heaviside function to reduce the equation to second-order ordinary differential equations with variable coefficients. We simplify these transformed equations using (i) the Laplace transformation technique in conjunction with convolution theory for solving moving force problems, and (ii) finite element analysis in conjunction with the Newmark method for solving analytically unsolvable moving mass problems due to their harmonic nature. We first solve the moving force problem using the finite element method and compare it against analytical solutions as validation for its accuracy in solving analytically unsolvable moving mass problems. The numerical solution obtained from the finite element method is shown to be comparable favorably against analytical solutions of our moving force problem. Lastly, we calculate displacement response curves for both moving distributed force and mass models at various time t for our dynamical problem presentation purposes.

Pollutant dispersion model in landfills by GILTT technique

The expansion of urban centers leads to an increase in the generation of municipal solid waste (MSW). In this way, due to the exorbitant amount of MSW produced annually, studies on the form of disposal of this waste are necessary, so that they are properly disposed of, due to environmental safety and reduction of risks to human health. Therefore, mathematical modeling is an essential tool in the study about the dispersion of pollutants in the porous medium, in this case, in relation to how the water table can be reached by these pollutants. Thus, this work intends to carry out a study of a two-dimensional model of the dispersion of pollutants in sanitary landfills, considering the governing equation in the dimensionless form. In the problem under study, a continuous and uniform leakage from a MSW storage cell is considered. The adopted model is solved by the Generalized Integral Laplace Transform Technique (GILTT) method, whose solution is analytical, except for the truncation error in the infinite series. The results obtained show that the proposed method is effective with the physical characteristics of the problem.

Hall effect on unsteady MHD flow along vertical plate with variable temperature and mass diffusion in the presence of porous medium and chemical reaction

The effects of chemical reaction and Hall current on the unsteady magneto hydrodynamic flow along a vertical plate with variable temperature and mass diffusion in the presence of a porous medium is studied hare. The fluid flow model of this paper contains governing partial differential equations of the motion, energy, and diffusion equation. To simplify the analysis, the governing equations are transformed into dimensionless form using non-dimensional variables. The Laplace-transform technique is employed to obtain an exact solution for the flow equations of the MHD model. The results of the analysis are presented using graphical representations of the velocity profiles. With the help of graphs, we discussed the behavior of fluid velocities with different parameters, including the chemical reaction parameter, Hall currents parameter, accelerated parameter, magnetic field parameter, and permeability parameter, and the numerical values of Sherwood number have been tabulated. We found that the values obtained for velocity, concentration and temperature are in concur-rence with the actual flow of the fluid

Magnetite/engine oil Casson nanofluid flow over a Riga plate with Atangana–Baleanu fractional derivative

The Riga plate is useful to enhance the conductivity of working fluids in many critical technical applications like MHD generators, nuclear reactor design, flow meters, and thermal nuclear reactors. The present work is carried out to analyze the heat transfer of ferrofluid with Engine oil as base fluid, flowing parallel to a vertical Riga plate. Atangana–Baleanu fractional derivative is used to model the unsteady distributions of velocity, temperature, and concentration fields, and exact solutions are acquired by applying the Laplace transformation technique and are exhibited with the help of special functions such as the Lorenzo–Hartley’s G function and the Mittag-Leffler function. The influence of all physical and rheological characteristics on flow distribution is illustrated graphically. From the study, it is revealed that the effect of modified Hartmann number plays a significant role in the velocity field and also that heat transfer is increased with increasing volume fraction of the nanoparticles. The effect of significant parameters on Sherwood number, Nusselt number, and skin friction is scrutinized in the tables.

Temperature Fluid Flow of Particles with The Effect of Magnetic Field in A Cylindrical Polar Coordinates of Incompressible Dusty Fluid

The effects of temperature of dusty fluid flow in a weak magnetic field are investigated in this paper. The motion of dusty fluid is affected by a homogeneous magnetic field that runs in the same direction as the fluid. Parameters include longitudinal velocity, Hartmann number; dust particle interactions, stock resistance, Reynolds number, and magnetic Reynolds number fully characterize the topic that is being studied. The obtained partial differential equations are translated into an ode system using appropriate similarity transformations. To solve these equations numerically we use, the Laplace Transform technique. The result shows the temperature of fluid particle reduced significantly in the magnetic field.

THERMAL DIFFUSION, AND THERMAL RADIATION EFFECTS ON UNSTEADY MHD CONVECTIVE FLOW PAST AN INCLINED VERTICAL PLATE

In this investigation, thermal diffusion and thermal radiation effects are studied on an unsteady natural convective flow of an electrically conducting fluid past an inclined vertical plate. A uniform aligned magnetic field is imposed onto the fluid or the plate. The governing PDEs are solved using the Laplace transformation technique. The effect of various parameters on velocity, temperature, concentration, skin friction, Nusselt number, and Sherwood number are analyzed graphically. It is observed that velocity field rises for increasing the value of the thermal-diffusion effect. Moreover, fluid concentration gets ososcillate with the effect of Prandtl number and thermal radiation parameter.

Application of q-HATM over Fractional in Time Model for Advection-Dispersion Equation with Variable Migration Parameters

In this article, an extended and more generalized method of homotopy analysis method (HAM), known as q-homotopy analysis transform method (q-HATM) is employed to develop solutions for highly nonlinear form of various time-fractional advection-dispersion models (TFADE). The proposed methodology of q-HATM is a conjunction of two different strong methodologies, that are HAM and Laplace transform technique (LTT) which makes the scheme much more capable to develop the convergent series solutions for differential equations with high nonlinearity. In this work, two different examples are constructed for the nonlinear time-fractional form of dispersion models corresponding to distance dependent dispersion and velocity components and also for the time dependent decay rate system and solutions are constructed by using q-HATM in generalized form. The graphical comparison is made for different types of variable functions and also for the different fractional order indexes. Prominent impact of fractional derivative indexes is significantly observed over both the examples of dispersion models. Hence, proposed examples shows that the q-HATM is much convenient to handle the nonlinear systems of fractional models.

MHD FLOW PAST OVER OSCILLATING VERTICAL PLATE THROUGH POROUS MEDIA IN THE PRESENCE OF HALL EFFECT AND CHEMICAL REACTION

MHD flow past over oscillating vertical plate with variable temperature and mass diffusion through a porous medium in the presence of Hall current effect and chemical reaction is studied here. The MHD fluid model contains partial differential equations of momentum, energy and diffusion equations. The Laplace-transform technique used to find the exact solution of equations involved in this paper. The resultant velocity profile is discussed with the help of graphs drawn for different parameters like, inclination of magnetic field, chemical reaction parameter, phase angle parameter and permeability parameter, and the numerical values Sherwood number have been also tabulated. It is noticed that the values obtained for velocity, concentration and temperature profiles are in concurrence with the actual flow of the unsteady MHD fluid.

Analysis of a diffusion-reaction heat transfer problem in a finite thickness layer adjoined by a semi-infinite medium

Multilayer diffusion-reaction problems are of much interest for both heat and mass transfer. While past work in this direction has mainly addressed multilayer bodies of finite size, practical problems such as immersion cooling of Li-ion cells and thermal runaway in semiconductor devices necessitate considering one of the layers to be semi-infinite. This work presents theoretical analysis of a diffusion-reaction problem in a finite layer surrounded by an infinite medium on both sides, where heat generation proportional to the local temperature occurs in the finite layer. Transient temperature distributions in the two bodies are determined using Laplace transformation technique. Through analysis of the poles of the solution in the Laplace domain, it is proved that this problem is unconditionally unstable, in that temperature in the finite thickness layer is predicted to always diverge at large times. However, the time taken to reach the unstable regime is shown to depend strongly on the key non-dimensional parameters of the problem. Temperature in the finite layer is shown to exhibit non-monotonic behavior at small times, particularly for large values of the heat generation coefficient, which is explained on the basis of the balance between temperature-dependent heat generation, heat dissipation into the semi-infinite medium and eventual slowdown of heat dissipation due to temperature rise. The impact of thermal properties on thermal behavior of the system is examined. A practical problem related to thermal safety design of a Li-ion cell is solved. Results from this work expand the state-of-the-art in theoretical analysis of diffusion-reaction problems, and also offer practical tools for thermal design of engineering problems including Li-ion cells and semiconductor devices.

Memory response on a piezoelectric rod under three-phase-lag model

ABSTRACT In this article, the concept of the memory-dependent derivative has been utilized to build up the heat equation to investigate the piezoelectric effect on a finite rod. The governing equations are written under the generalized theory of thermoelasticity based on the three-phase-lag model and solved in the Laplace transform domain. The numerical inverse Laplace transformation technique has been adapted to obtain the solution in the space–time domain. The effect of the different choices of kernel functions, time delay and comparison with other models are discussed and represented graphically. It is noted that the field variables attain the maximum amplitudes for the highest values of the time delay for a fixed kernel.

Hall and Ion Slip Impacts on Unsteady MHD Convective Flow of Ag-TiO2/WEG Hybrid Nanofluid in a Rotating Frame

Background: The radiative magnetohydrodynamic (MHD) flow of an incompressible viscous electrically conducting hybrid nanofluid over an exponentially accelerated vertical surface under the influence of slip velocity in a rotating frame taking Hall and ion slip impacts is discussed. Methods: Water and ethylene glycol mixture have been considered as a base fluid. A steady homogeneous magnetic field is applied under the assumption of a low magnetic Reynolds number. The ramped temperature and time-varying concentration at the surface are taken into consideration. The first-order consistent chemical reaction and heat absorption are also regarded. Silver (Ag) and titania (TiO2) nanoparticles are disseminated in base fluid water and ethylene glycol mixture to be formed a hybrid nanofluid. The Laplace transformation technique is employed on the non-dimensional governing equations for the closed form solutions. Results: The phrases for non-dimensional shear stresses, rates of heat, and mass transfer are also evaluated. The graphical representations are presented to scrutinize the effects of physical parameters on the significantflow characteristics. The computational values of the shear stresses, rates of heat and mass transports near the surface are tabulated by a range of implanted parameters. Conclusion: The resultant velocity grows by an increase in thermal and concentration buoyancy forces, Hall and ion-slip parameters, whereas rotation and slip parameters have overturn outcome on them. The temperature of hybrid Ag-TiO2/WEG nanofluid is relatively superior to that of Ag-WEG nanofluid. Species concentration of hybrid Ag-TiO2/WEG nanofluid decreases due to an increase in Schmidt number and chemical reaction parameter.

Numerical investigation of heat transfer & hall effects on MHD nanofluid flow past over an oscillating plate with radiation

The effects of convective heat generation and the oscillatory motion of a plate in the presence of MHD, Alumina nanofluid flow, thermal radiation, and Hall current are considered. The plate oscillates harmonically in its axes with uniform temperature. The dimensional equations have to be changed into non-dimensional equations with a set of dimensionless parameters. The Laplace transformation technique is utilized to get an exact solution. The possessions of velocity and temperature are analyzed with several parameters like Prandtl number (Pr), Grashof number (Gr), Hall parameter (m), magnetic parameter (M), radiation (R), solid volume fraction(ᵠ), phase angle(ω).The influence of primary and secondary velocity is discussed by the graph. It is observed that the increment of Hall parameter (m) diminishes the primary velocity, an increment of Grashof number leads to an increase in both velocities, and increasing solid volume fraction raises the temperature. The Nusselt number and skin friction coefficient values have expressed in the table. It is apparent that an increment of radiation increased the value of the Nusselt number and also an increment of phase angle value diminished the skin friction coefficient value.

Prabhakar fractional derivative model of sodium alginate (C6H9NaO7) for accelerated plate motions

The Prabhakar fractional derivative model is not studied in the open literature for the Casson fluid model when the vertical plate exhibits linear and quadratic translations with constant heating. Therefore, this study deals with the thermal transport of sodium alginate (C6H9NaO7) over a vertical plate with a constant temperature. Since the classical PDEs are incapable of analyzing and investigating the physical impact of flow variables with memory effects, a fractional derivative model is developed using the Prabhakar fractional derivative approach. Two different types of plate translations (linear and quadratic) are considered. The non-dimensional governing equations are transformed into a fractional model and solved using the Laplace transformation (L.T) technique. The effects and behavior of significant physical parameters and fractional order parameters are studied graphically and discussed. As a consequence, it is found that as fractional limitations are increased, the thermal and momentum profiles drop. In addition, the momentum profile in the case of quadratic translation (variable acceleration) shows a higher magnitude than the case of linear translation (constantly accelerated plate).

Chemical reaction impacts the unsteady MHD convective flow of an incompressible viscous fluid past an infinite vertical porous plate

AbstractWe studied the radiation magnetohydrodynamic flow of an incompressible viscous electrically conducting fluid past an exponentially accelerated perpendicular surface under the influence of slip velocity in the revolving structure. A steady homogeneous magnetic strength is applied under the assumption of low magnetic Reynolds quantity. The ramped heat and time‐altering concentration near the plate are taken into consideration. First‐order consistent chemical reactions and thermal absorption were also studied. The Laplace transformation technique is used for the non‐dimensional governing equations to get the closed‐form solutions. Supporting these results, the phases for nondimensional shear stress, rate of thermal as well as accumulation transport are also found. Graphical profiles are represented to examine the impact of physical parameters on the important physical flow features. The computational quantities of shear stress and rate of thermal and mass transportation near the surface are tabulated with a variety of implanted parameters. The resulting velocity is growing with an increase in heat and solutal buoyancy forces, while revolution and slip parameters have reverse effects on this. The resulting velocity is falling due to an increase in the Hartmann quantity, while the penetrability parameters have the opposite impacts on this. The species concentration of fluid is reduced by an increase in Schmidt number and chemical reacting parameter.

An analysis of heat source/sink influence on MHD fluid flow past a vertically enhanced plate with radiations

This paper explores the heat source/sink influence on irregular flow across a uniformly moving stretch sheet with of magnetic field. Flow equations are tackled with Laplace transformation technique. Tables and graphs drawn with MATLAB software show the effects of various physical parameters discovered during the numerical simulation on velocity, temperature, concentration, heat and mass transfer rate. The temperature is demonstrated to be reduced by the radiation and heat absorption parameters. Results reveal that concentration improves with enhancement of time. Furthermore, fluid velocity reduces with enhancement in viscosity. Solar energy collection systems, geophysics, astrophysics, aerospace and the design of high temperature manufacturing process systems are all examples of where the model can be used. The consistency of the model predictions is determined by comparing the findings to previously published literature. The article is unique and fills the gap in literature that the combined effects of heat source/sink with radiations and magnetism have rarely been researched.